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Bernstein functions and rates in mean ergodic theorems for operator semigroups. (English) Zbl 1306.47014

This paper deals with the rate of convergence in the mean ergodic theorem. More precisely, let \(T_t = e^{-tA}\) be a bounded, strongly continuous semigroup of operators on a Banach space \(X\), generated by \((A,D(A))\), and consider the Cesàro average \[ C_t(A) = \frac 1t\int_0^t T_sx\,ds. \] It is a standard fact that \(C_t(A)x\to y\) iff \(x\) is in the closure of the range of \(A\) and that \(C_t(A)x\to y\) implies that \(y\) is in the kernel of \(A\). This means that we have convergence of \(C_t(A)\) on the subspace \(\mathrm{ker}A\oplus\overline{\mathrm{range}A}\subset X\). The main result of the paper gives precise rates for this convergence depending on the properties of \(x\). This is achieved by a functional calculus.
In order to define \(g(A)\) for some function which is analytic on the right-half plane of \(\mathbb C\), the authors extend the classical Hille-Phillips calculus by using a regularization method (this is “construction two” of the automatic extensions in the sense of R. deLaubenfels [Stud. Math. 114, No. 3, 237–259 (1995; Zbl 0834.47012)]; the method is called “extended Hille-Phillips calculus” in [M. Haase, The functional calculus for sectorial operators. Operator Theory: Advances and Applications 169. Basel: Birkhäuser (2006; Zbl 1101.47010)]). If \(g\) is a Bernstein function (see [R. L. Schilling et al., Bernstein functions. Theory and applications. 2nd revised and extended ed. Berlin: de Gruyter (2012; Zbl 1257.33001)]), the authors show that \(g\) is admissible for the extended Hille-Phillips calculus and that the extended calculus coincides with the functional calculus induced by Bochner’s subordination (as in [R. L. Schilling, J. Aust. Math. Soc., Ser. A 64, No. 3, 368–396 (1998; Zbl 0920.47039)], see also [Zbl 1257.33001]). This allows them to formulate the main result:
Theorem. Let \(g\) be a Bernstein function such that \[ g(t) = a + bt + \int_{(0,\infty)} (1-e^{-st})\,\mu(ds) \] (the representing triplet \((a,b,\mu)\) uniquely determines \(g\); \(a,b\geq 0\) and \(\mu\) is a positive measure such that the integral converges) and set \[ r(t) = \frac 12 a + \frac 1t b + \int_{(0,\infty)} \frac st\wedge 1\,\mu(ds), \] then \(\mathrm{tr}(t)\) is a strictly positive, increasing and concave function. Moreover, there is always a Bernstein function \(g\) such that any \(\mathrm{tr}(t)\) which is strictly positive, increasing and convex function is connected, as above, with this Bernstein function via the representing triplet \((a,b,\mu)\). Under the assumption that \(g(0+)=0\), it is shown that, for all \(x\) in the domain of \(g(A)\),
(1)
\(\|C_t(A)g(A)x\|\leq 2Mr(t)\|x\|\), where \(M = \sup_s\|T_s\|\);
(2)
\(\|C_t(A)g(A)x\| = o(r(t))\) as \(t\to\infty\), provided that \((T_s)_s\) is mean ergodic and \(\lim_{t\to\infty} \mathrm{tr}(t)=\infty\);
(3)
\(\|C_t(A)g(A)x\| = O(1/t)\) as \(t\to\infty\) whenever \(\mathrm{tr}(t)=o(1)\) as \(t\to\infty\).
In particular, one can obtain any convergence rate by a suitable choice of \(g\). If \(f\) is a function such that \(1/f\) is a Bernstein function (e.g., a potential or a Stieltjes function) then the main result can be transferred as follows: Theorem. Assume that \[ f(0+)=\infty \quad\text{and}\quad \text{w-}\lim_{s\to 0}\int_0^\infty e^{-as}T_sx\,\mu(ds)=y, \] then
(1)
for \(x= \frac 1f(A)y\) one has \(\|C_t(A)x\|=O(1/f(1/t))\) as \(t\to\infty\);
(2)
\(\|C_t(A)x\|=o(1/f(1/t))\) provided that \(\lim_{t\to\infty} t/f(1/t)=\infty\).
The paper ends with the discussion of a few important examples.

MSC:

47A35 Ergodic theory of linear operators
47A40 Scattering theory of linear operators
47D03 Groups and semigroups of linear operators
47A60 Functional calculus for linear operators
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